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Large Scale Analytic Calculations in Quantum Field Theories

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Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

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Abstract

We present a survey on the mathematical structure of zero- and single scale quantities and the associated calculation methods and function spaces in higher order perturbative calculations in relativistic renormalizable quantum field theories.

Dedicated to Peter Paule on the occasion of his 60th birthday

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Notes

  1. 1.

    For other recent surveys on integration methods for Feynman integrals see [6,7,8,9,10,11].

  2. 2.

    Correspondingly, in the case of more parameters, partial differential equation systems are obtained.

  3. 3.

    This representation has been used in a more special form also in [173] later.

  4. 4.

    For a recent application to the one-loop case, see e.g. [237].

  5. 5.

    Jacobi to Legendre, July 2nd, 1830.

References

  1. The ILC: https://en.wikipedia.org/wiki/International_Linear_Collider Aguilar-Saavedra, J.A., et al., [ECFA/DESY LC Physics Working Group]: TESLA: The Superconducting electron positron linear collider with an integrated x-ray laser laboratory. Technical design report. Part 3. Physics at an e +e − linear collider, hep-ph/0106315; Accomando, E., et al., [ECFA/DESY LC Physics Working Group]: Phys. Rept. 299, 1–78 (1998). [hep-ph/9705442]

  2. The Future Circular Collider. https://en.wikipedia.org/wikiFuture_Circular_Collider. TH FCC-ee design study. http://tlep.web.cern.ch

  3. ’t Hooft, G., Veltman, M.J.G.: Nucl. Phys. B153, 365–401 (1979)

    Google Scholar 

  4. ’t Hooft, G., Veltman, M.J.G.: NATO Sci. Ser. B 4, 177–322 (1974)

    Google Scholar 

  5. Veltman, M.J.G.: Diagrammatica: The Path to Feynman Rules. Cambridge Lecture Notes in Physics, vol. 4. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  6. Weinzierl, S.: Introduction to Feynman Integrals (2010). arXiv:1005.1855 [hep-ph]

    Google Scholar 

  7. Ablinger, J., Blümlein, J., Schneider, C.: J. Phys. Conf. Ser. 523, 012060 (2014). [arXiv:1310.5645[math-ph]]. http://www.arXiv.org/abs/[arXiv:1310.5645[math-ph]]

    Article  Google Scholar 

  8. Ablinger, J., Blümlein, J.: In: Schneider C., Blümlein J. (eds.) Computer Algebra in Quantum Field Theory. Integration, Summation and Special Functions, pp. 1–32. Springer, Wien (2012). [arXiv: 1304.7071 [math-ph]]

    Google Scholar 

  9. Weinzierl, S.: In: Schneider C., Blümlein J. (eds.) Computer Algebra in Quantum Field Theory. Integration, Summation and Special Functions, pp. 381–406. Springer, Wien (2012). [arXiv:1301.6918 [hep-ph]]

    Google Scholar 

  10. Duhr, C.: In: Dixon L., Petriello F. (eds.) Journeys Through the Precision Frontier: Amplitudes for Colliders, 2014 TASI Lectures, pp. 419–476. World Scientific, Singapore (2015). [arXiv:1411.7538 [hep-ph]]

    Google Scholar 

  11. Blümlein, J., Schneider, C.: Int. J. Mod. Phys. A 33(17), 1830015 (2018). [arXiv:1809.02889 [hep-ph]]

    Google Scholar 

  12. Nogueira, P.: J. Comput. Phys. 105, 279–289 (1993)

    Article  MathSciNet  Google Scholar 

  13. van Ritbergen, T., Schellekens, A., Vermaseren, J.A.M.: Int. J. Mod. Phys. A14, 41–96 (1999). [hep-ph/9802376]. http://www.arXiv.org/abs/[hep-ph/9802376]

    Google Scholar 

  14. Nakanishi, N.: Graph Theory and Feynman Integrals. Gordon and Breach, New York (1971)

    MATH  Google Scholar 

  15. Lefschetz, S.: Applications of Algebraic Topology: Graphs and Networks, the Picard-Lefschetz Theory an Feynman Integrals. Springer, Berlin (1975)

    Book  MATH  Google Scholar 

  16. Bogner, C., Weinzierl, S.: Int. J. Mod. Phys. A25, 2585–2618 (2010). [arXiv:1002.3458[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1002.3458[hep-ph]]

    Article  Google Scholar 

  17. Vermaseren, J.A.M.: New features of FORM (2000). math-ph/0010025

    Google Scholar 

  18. Tentyukov, M., Fliegner, D., Frank, M., Onischenko, A., Retey, A., Staudenmaier, H.M., Vermaseren, J.A.M.: AIP Conf. Proc. 583(1), 202 (2002). [cs/0407066 [cs-sc]]

    Google Scholar 

  19. Tentyukov, M., Vermaseren, J.A.M.: Comput. Phys. Commun. 181, 1419–1427 (2010). [hep-ph/0702279]. http://www.arXiv.org/abs/[hep-ph/0702279]

    Google Scholar 

  20. Ruijl, B., Takahiro, U., Vermaseren, J.A.M.: FORM version 4.2 (2017). arXiv: 1707.06453[hep-ph]

    Google Scholar 

  21. Chetyrkin, K.G., Tkachov, F.V.: Nucl. Phys. B192, 159–204 (1981)

    Article  Google Scholar 

  22. Laporta, S.: Int. J. Mod. Phys. A15, 5087–5159 (2000). [hep-ph/0102033]. http://www.arXiv.org/abs/[hep-ph/0102033]

    Google Scholar 

  23. Smirnov, A.: JHEP 10, 107 (2008). [arXiv:0807.3243[hep-ph]]. http://www.arXiv.org/abs/[arXiv:0807.3243[hep-ph]]

    Article  Google Scholar 

  24. Smirnov, A.V., Chuharev, F.S.: FIRE6: Feynman Integral REduction with Modular Arithmetic (2019). arXiv:1901.07808 [hep-ph]

    Google Scholar 

  25. Studerus, C.: Comput. Phys. Commun. 181, 1293–1300 (2010). [arXiv:0912.2546 [physics.comp-ph]]. http://www.arXiv.org/abs/[arXiv:0912.2546[physics.comp-ph]]

    Google Scholar 

  26. von Manteuffel, A., Studerus, C.: Reduze 2 – Distributed Feynman Integral Reduction (2012). arXiv:1201.4330 [hep-ph]

    Google Scholar 

  27. Marquard, P., Seidel, D.: The Crusher algorithm (unpublished)

    Google Scholar 

  28. Gerhold, S.: Uncoupling systems of linear Ore operator equations. Master’s thesis, RISC, J. Kepler University, Linz (2002)

    Google Scholar 

  29. Bostan, A., Chyzak, F., de Panafieu, É.: Complexity estimates for two uncoupling algorithms. In: Proceedings of ISSAC’13, Boston, June (2013)

    Google Scholar 

  30. Zürcher, B.: Rationale Normalformen von pseudo-linearen Abbildungen. Master’s thesis, Mathematik, ETH Zürich (1994)

    Google Scholar 

  31. Karr, M.: J. ACM 28, 305–350 (1981)

    Article  MathSciNet  Google Scholar 

  32. Bronstein, M.: J. Symbolic Comput. 29(6), 841–877 (2000)

    Google Scholar 

  33. Schneider, C.: Symbolic summation in difference fields. Ph.D. Thesis RISC, Johannes Kepler University, Linz technical report 01–17 (2001)

    Google Scholar 

  34. Schneider, C.: An. Univ. Timisoara Ser. Mat.-Inform. 42, 163–179 (2004)

    Google Scholar 

  35. Schneider, C.: J. Differ. Equations Appl. 11, 799–821 (2005)

    Article  Google Scholar 

  36. Schneider, C.: Appl. Algebra Eng. Commun. Comput. 16, 1–32 (2005)

    Article  Google Scholar 

  37. Schneider, C.: J. Algebra Appl. 6, 415–441 (2007)

    Article  MathSciNet  Google Scholar 

  38. Schneider, C.: Clay Math. Proc. 12, 285–308 (2010). [arXiv:0904.2323 [cs.SC]]. [arXiv:0904.2323]

    Google Scholar 

  39. Schneider, C.: Ann. Comb. 14, 533–552 (2010). [arXiv:0808.2596]

    Article  MathSciNet  Google Scholar 

  40. Schneider, C.: In: Gutierrez J., Schicho J., Weimann M. (eds.) Computer Algebra and Polynomials, Applications of Algebra and Number Theory. Lecture Notes in Computer Science (LNCS), vol. 8942, pp. 157–191 (2015). [arXiv:1307.7887 [cs.SC]]

    Google Scholar 

  41. Schneider, C.: J. Symbolic Comput. 43, 611–644 (2008). [arXiv:0808.2543]

    Article  MathSciNet  Google Scholar 

  42. Schneider, C.: J. Symb. Comput. 72, 82–127 (2016). [arXiv:1408.2776 [cs.SC]]

    Google Scholar 

  43. Schneider, C.: J. Symb. Comput. 80, 616–664 (2017). [arXiv:1603.04285 [cs.SC]]

    Google Scholar 

  44. Schneider, C.: Sém. Lothar. Combin. 56, 1–36 (2007). article B56b. http://www.arXiv.org/abs/articleB56b

  45. Schneider, C.: In: Schneider C., Blümlein J. (eds.) Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions. Texts and Monographs in Symbolic Computation, pp. 325–360. Springer, Wien (2013). [arXiv:1304.4134 [cs.SC]]

    Google Scholar 

  46. Binoth, T., Heinrich, G.: Nucl. Phys. B585, 741–759 (2000). [hep-ph/0004013]. http://www.arXiv.org/abs/[hep-ph/0004013]

    Google Scholar 

  47. Nagy, Z., Soper, D.E.: Phys. Rev. D74, 093006 (2006). [hep-ph/0610028]. http://www.arXiv.org/abs/[hep-ph/0610028]

    Google Scholar 

  48. Anastasiou, C., Beerli, S., Daleo, A.: JHEP 05, 071 (2007). [hep-ph/0703282]. http://www.arXiv.org/abs/[hep-ph/0703282]

    Article  Google Scholar 

  49. Smirnov, A.V., Tentyukov, M.N.: Comput. Phys. Commun. 180, 735–746 (2009). [arXiv:0807.4129[hep-ph]]. http://www.arXiv.org/abs/[arXiv:0807.4129[hep-ph]]

    Article  Google Scholar 

  50. Carter, J., Heinrich, G.: Comput. Phys. Commun. 182, 1566–1581 (2011). [arXiv:1011.5493[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1011.5493[hep-ph]]

    Article  Google Scholar 

  51. Smirnov, A.V., Smirnov, V.A., Tentyukov, M.: Comput. Phys. Commun. 182, 790–803 (2011). [arXiv:0912.0158[hep-ph]]. http://www.arXiv.org/abs/[arXiv:0912.0158[hep-ph]]

    Article  Google Scholar 

  52. Becker, S., Reuschle, C., Weinzierl, S.: JHEP 12, 013 (2010). [arXiv:1010.4187[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1010.4187[hep-ph]]

    Article  Google Scholar 

  53. Becker, S., Reuschle, C., Weinzierl, S.: JHEP 1207, 090 (2012). [arXiv:1205.2096[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1205.2096[hep-ph]]

    Article  Google Scholar 

  54. Becker, S., Götz, D., Reuschle, C., Schwan, C., Weinzierl, S.: Phys. Rev. Lett. 108, 032005 (2012). [arXiv:1111.1733[hel-ph]]. http://www.arXiv.org/abs/[arXiv:1111.1733[hel-ph]]

    Article  Google Scholar 

  55. Smirnov, A.V.: Comput. Phys. Commun. 204, 189–199 (2016). [arXiv:1511.03614[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1511.03614[hep-ph]]

    Article  Google Scholar 

  56. Borowka, S., Gehrmann, T., Hulme, D.: JHEP 1808, 111 (2018). [arXiv:1804.06824 [hep-ph]]

    Article  Google Scholar 

  57. Ablinger, J., Blümlein, J., Klein, S., Schneider, C.: Nucl. Phys. Proc. Suppl. 205–206, 110–115 (2010). [arXiv:1006.4797[math-ph]]. http://www.arXiv.org/abs/[arXiv:1006.4797[math-ph]]

    Article  Google Scholar 

  58. Blümlein, J., Hasselhuhn, A., Schneider, C.: PoS (RADCOR2011), 032 (2012). [arXiv: 1202.4303[math-ph]]. http://www.arXiv.org/abs/[arXiv:1202.4303[math-ph]]

    Google Scholar 

  59. Schneider, C.: J. Phys. Conf. Ser. 523, 012037 (2014). [arXiv:1310.0160[cs.Sc]]. http://www.arXiv.org/abs/[arXiv:1310.0160[cs.Sc]]

    Article  Google Scholar 

  60. Krattenthaler, C., Schneider, C.: Evaluation of binomial double sums involving absolute values. In: Pillwein, V., Schneider, C. (eds) Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra. Texts & Monographs in Symbolic Computation. Springer, Heidelberg (2020). https://doi.org/10.1007/978-3-030-44559-1_14

    Google Scholar 

  61. Ferguson, H.R.P., Bailey, D.H.: A polynomial time, numerically stable integer relation algorithm, RNR Technical Report RNR-91-032, July 14 (1992)

    Google Scholar 

  62. Luthe, T., Maier, A., Marquard, P., Schröder, Y.: JHEP 10, 166 (2017). [arXiv: 1709.07718[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1709.07718[hep-ph]]

    Article  Google Scholar 

  63. Borwein, J.M., Bradley, D.M., Broadhurst, D.J., Lisonek, P.: Trans. Am. Math. Soc. 353, 907–941 (2001). [math/9910045]. http://www.arXiv.org/abs/[math/9910045]

    Google Scholar 

  64. Blümlein, J., Broadhurst, D., Vermaseren, J.A.M.: Comput. Phys. Commun. 181, 582–625 (2010). [arXiv: 0907.2557[math-ph]]. http://www.arXiv.org/abs/[arXiv:0907.2557[math-ph]]

    Google Scholar 

  65. Klein, F.: Vorlesungen über die hypergeometrische Funktion. Wintersemester 1893/94, Die Grundlehren der Mathematischen Wissenschaften, vol. 39. Springer, Berlin (1933)

    Google Scholar 

  66. Bailey, W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935)

    MATH  Google Scholar 

  67. Slater, L.J.: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge (1966)

    MATH  Google Scholar 

  68. Appell, P., Kampé de Fériet, J.: Fonctions Hypergéométriques et Hypersphériques, Polynomes D’ Hermite. Gauthier-Villars, Paris (1926)

    Google Scholar 

  69. Appell, P.: Les Fonctions Hypergëométriques de Plusieur Variables. Gauthier-Villars, Paris (1925)

    Google Scholar 

  70. Kampé de Fériet, J.: La fonction hypergëométrique. Gauthier-Villars, Paris (1937)

    Google Scholar 

  71. Exton, H.: Multiple Hypergeometric Functions and Applications. Ellis Horwood, Chichester (1976)

    MATH  Google Scholar 

  72. Exton, H.: Handbook of Hypergeometric Integrals. Ellis Horwood, Chichester (1978)

    MATH  Google Scholar 

  73. Schlosser, M.J.: In: Schneider C., Blümlein J. (eds.) Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, pp. 305–324. Springer, Wien (2013). [arXiv:1305.1966 [math.CA]]

    Google Scholar 

  74. Anastasiou, C., Glover, E.W.N., Oleari, C.: Nucl. Phys. B572, 307–360 (2000). [hep-ph/9907494]. http://www.arXiv.org/abs/[hep-ph/9907494]

    Google Scholar 

  75. Anastasiou, C., Glover, E.W.N., Oleari, C.: Nucl. Phys. B565, 445–467 (2000). [hep-ph/9907523]. http://www.arXiv.org/abs/[hep-ph/9907523]

    Google Scholar 

  76. Srivastava, H.M., Karlsson, P.W.: Multiple Gaussian Hypergeometric Series. Ellis Horwood, Chichester (1985)

    MATH  Google Scholar 

  77. Lauricella, G.: Rediconti del Circolo Matematico di Palermo 7(S1), 111–158 (1893)

    Google Scholar 

  78. Saran, S.: Ganita 5, 77–91 (1954)

    Google Scholar 

  79. Saran, S.: Acta Math. 93, 293–312 (1955)

    Article  MathSciNet  Google Scholar 

  80. Hamberg, R., van Neerven, W.L., Matsuura, T.: Nucl. Phys. B359, 343–405 (1991). [Erratum: Nucl. Phys. B644 (2002) 403–404]

    Google Scholar 

  81. Hamberg, R.: Second order gluconic contributions to physical quantities. Ph.D. Thesis, Leiden University (1991)

    Google Scholar 

  82. Buza, M., Matiounine, Y., Smith, J., Migneron, R., van Neerven, W.L.: Nucl. Phys. B472, 611–658 (1996). [hep-ph/9601302]. http://www.arXiv.org/abs/[hep-ph/9601302]

    Google Scholar 

  83. Bierenbaum, I., Blümlein, J., Klein, S.: Nucl. Phys. B780, 40–75 (2007). [hep-ph/0703285]. http://www.arXiv.org/abs/[hep-ph/0703285]

    Google Scholar 

  84. Ablinger, J., Blümlein, J., Hasselhuhn, A., Klein, S., Schneider, C., Wißbrock, F.: Nucl. Phys. B864, 52–84 (2012). [arXiv:1206.2252[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1206.2252[hep-ph]]

    Article  Google Scholar 

  85. Ablinger, J., Behring, A., Blümlein, J., De Freitas, A., von Manteuffel, A., Schneider, C.: Comput. Phys. Commun. 202, 33–112 (2016). [arXiv:1509.08324[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1509.08324[hep-ph]]

    Article  MathSciNet  Google Scholar 

  86. Pochhammer, L.: Math. Ann. 35, 495–526 (1890)

    Article  MathSciNet  Google Scholar 

  87. Kratzer, A., Franz, W.: Transzendente Funktionen. Geest & Portig, Leipzig (1960)

    MATH  Google Scholar 

  88. Barnes, E.W.: Q. J. Math. 41, 136–140 (1910)

    Google Scholar 

  89. Mellin, H.: Math. Ann. 68(3), 305–337 (1910)

    Google Scholar 

  90. Smirnov, V.A.: Feynman Integral Calculus. Springer, Berlin (2006)

    Google Scholar 

  91. Czakon, M.: Comput. Phys. Commun. 175, 559–571 (2006). [hep-ph/0511200]. http://www.arXiv.org/abs/[hep-ph/0511200]

    Google Scholar 

  92. Smirnov, A., Smirnov, V.: Eur. Phys. J. C62, 445–449 (2009). [arXiv: 0901.0386[hep-ph]]. http://www.arXiv.org/abs/[arXiv:0901.0386[hep-ph]]

    Google Scholar 

  93. Gluza, J., Kajda, K., Riemann, T.: Comput. Phys. Commun. 177, 879–893 (2007). [arXiv:0704.2423[hep-ph]]. http://www.arXiv.org/abs/[arXiv:0704.2423[hep-ph]]

    Article  Google Scholar 

  94. Gluza, J., Kajda, K., Riemann, T., Yundin, V.: Eur. Phys. J. C71, 1516 (2011). [arXiv:1010.1667[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1010.1667[hep-ph]]

    Article  Google Scholar 

  95. Ablinger, J., Blümlein, J., De Freitas, A., Schneider, C., Schönwald, K.: Nucl. Phys. B927, 339–367 (2018). [arXiv: 1711.06717[hep-th]]. http://www.arXiv.org/abs/[arXiv:1711.06717[hep-th]]

    Google Scholar 

  96. Kummer, E.E.: J. Reine Angew. Math. (Crelle) 21, 74–90; 193–225; 328–371 (1840)

    Google Scholar 

  97. Poincaré, H.: Acta Math. 4, 201–312 (1884)

    Article  MathSciNet  Google Scholar 

  98. Lappo-Danilevsky, J.A.: Mémoirs sur la Théorie des Systèmes Différentielles Linéaires, Chelsea Publ. Co, New York (1953)

    MATH  Google Scholar 

  99. Chen, K.T: Trans. Am. Math. Soc. 156(3), 359–379 (1971)

    Google Scholar 

  100. Goncharov, A.B.: Math. Res. Lett. 5, 497–516 (1998)

    Article  MathSciNet  Google Scholar 

  101. von Manteuffel, A., Panzer, E., Schabinger, R.M.: JHEP 02, 120 (2015). [arXiv:1411.7392[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1411.7392[hep-ph]]

    Article  Google Scholar 

  102. Brown, F.: Commun. Math. Phys. 287, 925–958 (2009). [arXiv:0804.1660 [math.AG]]

    Google Scholar 

  103. Panzer, E.: Comput. Phys. Commun. 188, 148–166 (2015). [arXiv:1403.3385[hep-th]]. http://www.arXiv.org/abs/[arXiv:1403.3385[hep-th]]

    Article  Google Scholar 

  104. Ablinger, J., Blümlein, J., Raab, C., Schneider, C., Wißbrock, F.: Nucl. Phys. B885, 409–447 (2014). [arXiv:1403.1137[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1403.1137[hep-ph]]

    Article  Google Scholar 

  105. Wißbrock, F.Ph.: \(O(\alpha _s^3)\) contributions to the heavy flavor Wilson coefficients of the structure function F2(x, Q 2) at Q 2 ≫ m 2. Ph.D. Thesis, TU Dortmund (2015)

    Google Scholar 

  106. Kotikov, A.V.: Phys. Lett. B254, 158–164 (1991)

    Article  Google Scholar 

  107. Bern, Z., Dixon, L.J., Kosower, D.A.: Phys. Lett. B302, 299–308 (1993). [Erratum: Phys. Lett. B318, (1993) 649]. [hep-ph/9212308]. http://www.arXiv.org/abs/[hep-ph/9212308]

    Google Scholar 

  108. Remiddi, E.: Nuovo Cim. A110, 1435–1452 (1997). [hep-th/9711188]. http://www.arXiv.org/abs/[hep-th/9711188]

    Google Scholar 

  109. Gehrmann, T., Remiddi, E.: Nucl. Phys. B580, 485–518 (2000). [hep-ph/9912329]. http://www.arXiv.org/abs/[hep-ph/9912329]

    Google Scholar 

  110. Abramov, S.A., Petkovšek, M.: D’Alembertian solutions of linear differential and difference equations. In: von zur Gathen J. (ed.) Proceedings of ISSAC’94, pp. 169–174. ACM Press, New York (1994)

    Google Scholar 

  111. Ablinger, J.: PoS (RADCOR2017), 001 (2017). [arXiv:1801.01039 [cs.SC]]

    Google Scholar 

  112. Singer, M.F.: Am. J. Math. 103(4), 661–682 (1981)

    Google Scholar 

  113. Kovacic, J.J.: J. Symb. Comput. 2, 3–43 (1986)

    Article  Google Scholar 

  114. Blümlein, J., Ablinger, J., Behring, A., De Freitas, A., von Manteuffel, A., Schneider, C.: PoS (QCDEV2017), 031 (2017). [arXiv:1711.07957 [hep-ph]]

    Google Scholar 

  115. Ablinger, J., Behring, A., Blümlein, J., Falcioni, G., De Freitas, A., Marquard, P., Rana, N., Schneider, C.: Phys. Rev. D 97(9), 094022 (2018). [arXiv:1712.09889 [hep-ph]]

    Google Scholar 

  116. Ablinger, J., Blümlein, J., Marquard, P., Rana, N., Schneider, C.: Nucl. Phys. B 939, 253–291 (2019). [arXiv:1810.12261 [hep-ph]]

    Google Scholar 

  117. Kotikov, A.V.: The property of maximal transcendentality in the N=4 supersymmetric Yang-Mills. In: Diakonov D. (ed.) Subtleties in quantum field theory, pp. 150–174. [arXiv:1005.5029 [hep-th]]

    Google Scholar 

  118. Henn, J.M.: Phys. Rev. Lett. 110, 251601 (2013). [arXiv:1304.1806[hep-th]]. http://www.arXiv.org/abs/[arXiv:1304.1806[hep-th]]

    Article  Google Scholar 

  119. Henn, J.M.: J. Phys. A48, 153001 (2015). [arXiv:1412.2296[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1412.2296[hep-ph]]

    Google Scholar 

  120. Zakharov, V.E., Manakov, S.V., Novikov, S.P., Pitaevskii, L.P.: Teoria Solitonov: metod obratnoi zadatschi. Nauka, Moskva (1980)

    Google Scholar 

  121. Sakovich, S.Yu.: J. Phys. A Math. Gen. 28, 2861–2869 (1995)

    Article  Google Scholar 

  122. Lee, R.N.: JHEP 04, 108 (2015). [arXiv:1411.0911[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1411.0911[hep-ph]]

    Article  Google Scholar 

  123. Prausa, M.: Comput. Phys. Commun. 219, 361–376 (2017). [arXiv:1701.00725 [hep-ph]]. http://www.arXiv.org/abs/[arXiv:1701.00725[hep-ph]]

    Google Scholar 

  124. Gituliar, O., Magerya, V.: Comput. Phys. Commun. 219, 329–338 (2017). [arXiv:1701.04269[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1701.04269[hep-ph]]

    Article  MathSciNet  Google Scholar 

  125. Meyer, C.: Comput. Phys. Commun. 222, 295–312 (2018). [arXiv:1705.06252[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1705.06252[hep-ph]]

    Article  Google Scholar 

  126. Ablinger, J., Behring, A., Blümlein, J., De Freitas, A., Hasselhuhn, A., von Manteuffel, A., Round, M., Schneider, C., Wißbrock, F.: Nucl. Phys. B886, 733–823 (2014). [arXiv:1406.4654[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1406.4654[hep-ph]]

    Article  Google Scholar 

  127. Ablinger, J., Behring, A., Blümlein, J., De Freitas, A., von Manteuffel, A., Schneider, C.: Nucl. Phys. B890, 48–151 (2014). [arXiv:1409.1135[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1409.1135[hep-ph]]

    Google Scholar 

  128. Henn, J., Smirnov, A.V., Smirnov, V.A., Steinhauser, M.: JHEP 01, 074 (2017). [arXiv:1611.07535[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1611.07535[hep-ph]]

    Article  Google Scholar 

  129. Kauers, M., Paule, P.: The concrete tetrahedron. In: Texts and Monographs in Symbolic Computation. Springer, Wien (2011)

    Google Scholar 

  130. Blümlein, J., Schneider, C.: Phys. Lett. B771, 31–36 (2017). [arXiv: 1701.04614 [hep-ph]]. http://www.arXiv.org/abs/[arXiv:1701.04614[hep-ph]]

    Google Scholar 

  131. Gorishnii, S.G., Larin, S.A., Surguladze, L.R., Tkachov, F.V., Comput. Phys. Commun. 55, 381–408 (1989). Larin, S.A., Tkachov, F.V., Vermaseren, J.A.M.: The FORM version of MINCER, NIKHEF-H-91-18

    Google Scholar 

  132. Steinhauser, M.: Comput. Phys. Commun. 134, 335–364 (2001). [hep-ph/ 0009029]

    Google Scholar 

  133. Harlander, R., Seidensticker, T., Steinhauser, M.: Phys. Lett. B426, 125–132 (1998). [hep-ph/9712228]. http://www.arXiv.org/abs/[hep-ph/9712228]

    Google Scholar 

  134. Seidensticker, T.: Automatic application of successive asymptotic expansions of Feynman diagrams. In: Proceedings of the 6th International Workshop on New Computing Techniques in Physics Research, Crete, April (1999). hep-ph/9905298

    Google Scholar 

  135. Nörlund, N.E.: Vorlesungen über Differenzenrechnung. Springer, Berlin (1924). Reprinted by (Chelsea Publishing Company, New York, 1954)

    Google Scholar 

  136. Larin, S.A., van Ritbergen, T., Vermaseren, J.A.M.: Nucl. Phys. B427, 41–52 (1994)

    Article  Google Scholar 

  137. Larin, S.A., Nogueira, P., van Ritbergen, T., Vermaseren, J.A.M.: Nucl. Phys. B492, 338–378 (1997). [hep-ph/9605317]. http://www.arXiv.org/abs/[hep-ph/9605317]

    Google Scholar 

  138. Retey, A., Vermaseren, J.A.M.: Nucl. Phys. B604, 281–311 (2001). [hep-ph/0007294]. http://www.arXiv.org/abs/[hep-ph/0007294]

    Google Scholar 

  139. Blümlein, J., Vermaseren, J.A.M.: Phys. Lett. B606, 130–138 (2005). [hep-ph/0411111]. http://www.arXiv.org/abs/[hep-ph/0411111]

    Google Scholar 

  140. Bierenbaum, I., Blümlein, J., Klein, S.: Nucl. Phys. B820, 417–482 (2009). [arXiv:0904.3563[hep-ph]]. http://www.arXiv.org/abs/[arXiv:0904.3563[hep-ph]]

    Article  Google Scholar 

  141. Vermaseren, J.A.M., Vogt, A., Moch, S.: Nucl. Phys. B724, 3–182 (2005). [hep-ph/0504242]. http://www.arXiv.org/abs/[hep-ph/0504242]

    Google Scholar 

  142. Kauers, M., Jaroschek, M., Johansson, F.: In: Gutierrez J., Schicho J., Weimann M. (eds.) Computer Algebra and Polynomials. Lecture Notes in Computer Science, vol. 8942, pp. 105–125. Springer, Berlin (2015). [arXiv:1306.4263 [cs.SC]]

    Google Scholar 

  143. Sage. http://www.sagemath.org/

  144. Blümlein, J., Kauers, M., Klein, S., Schneider, C.: Comput. Phys. Commun. 180, 2143–2165 (2009). [arXiv:0902.4091[hep-ph]]. http://www.arXiv.org/abs/[arXiv:0902.4091[hep-ph]]

    Article  Google Scholar 

  145. Moch, S., Vermaseren, J.A.M., Vogt, A.: Nucl. Phys. B688, 101–134 (2004). [hep-ph/0403192]. http://www.arXiv.org/abs/[hep-ph/0403192]

    Google Scholar 

  146. Vogt, A., Moch, S., Vermaseren, J.A.M.: Nucl. Phys. B691, 129–181 (2004). [hep-ph/0404111]. http://www.arXiv.org/abs/[hep-ph/0404111]

    Google Scholar 

  147. Ablinger, J., Behring, A., Blümlein, J., De Freitas, A., von Manteuffel, A., Schneider, C.: Nucl. Phys. B922, 1–40 (2017). [arxiv:1705.01508[hep-ph]]. http://www.arXiv.org/abs/[arxiv:1705.01508[hep-ph]]

    Article  Google Scholar 

  148. Ablinger, J., Blümlein, J., Marquard, P., Rana, N., Schneider, C.: Phys. Lett. B 782, 528–532 (2018). [arXiv:1804.07313 [hep-ph]]

    Google Scholar 

  149. Almkvist, G., Zeilberger, D.: J. Symb. Comput. 10, 571–591 (1990)

    Article  Google Scholar 

  150. Apagodu, M., Zeilberger, D.: Adv. Appl. Math. (Special Regev Issue) 37, 139–152 (2006)

    Google Scholar 

  151. Ablinger, J.: Computer Algebra Algorithms for Special Functions in Particle Physics. Ph.D. Thesis, Linz U. (2012). arXiv:1305.0687[math-ph]

    Google Scholar 

  152. Broadhurst, D.J., Fleischer, J., Tarasov, O.V.: Z. Phys. C60, 287–302 (1993). [hep-ph/9304303]. http://www.arXiv.org/abs/[hep-ph/9304303]

    Google Scholar 

  153. Bloch, S., Vanhove, P.: J. Number Theor. 148, 328–364 (2015). [hep-th/ 1309.5865]. http://www.arXiv.org/abs/[hep-th/1309.5865]

    Google Scholar 

  154. Laporta, S., Remiddi, E.: Nucl. Phys. B704, 349–386 (2005). [hep-ph/0406160]. http://www.arXiv.org/abs/[hep-ph/0406160]

    Google Scholar 

  155. Adams, L., Bogner, C., Weinzierl, S.: J. Math. Phys. 54, 052303 (2013). [hep-ph/1302.7004]. http://www.arXiv.org/abs/[hep-ph/1302.7004]

    Article  MathSciNet  Google Scholar 

  156. Adams, L., Bogner, C., Weinzierl, S.: J. Math. Phys. 55(10), 102301 (2014). [hep-ph/1405.5640]. http://www.arXiv.org/abs/[hep-ph/1405.5640]

    Google Scholar 

  157. Adams, L., Bogner, C., Weinzierl, S.: J. Math. Phys. 56(7), 072303 (2015). [hep-ph/1504.03255]. http://www.arXiv.org/abs/[hep-ph/1504.03255]

    Google Scholar 

  158. Adams, L., Bogner, C., Weinzierl, S.: J. Math. Phys. 57(3), 032304 (2016). [hep-ph/1512.05630]. http://www.arXiv.org/abs/[hep-ph/1512.05630]

    Google Scholar 

  159. Sabry, A.: Nucl. Phys. 33, 401–430 (1962)

    Article  Google Scholar 

  160. Remiddi, E., Tancredi, L.: Nucl. Phys. B907, 400–444 (2016). [arXiv:1602.01481[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1602.01481[hep-ph]]

    Article  Google Scholar 

  161. Adams, L., Bogner, C., Schweitzer, A., Weinzierl, S.: J. Math. Phys. 57(12), 122302 (2016). [hep-ph/1607.01571]. http://www.arXiv.org/abs/[hep-ph/1607.01571]

    Google Scholar 

  162. Ablinger, J., Blümlein, J., De Freitas, A., van Hoeij, M., Imamoglu, E., Raab, C.G., Radu, C.S., Schneider, C.: J. Math. Phys. 59(6), 062305 (2018). [arXiv: 1706.01299 [hep-th]]

    Google Scholar 

  163. Grigo, J., Hoff, J., Marquard, P., Steinhauser, M.: Nucl. Phys. B 864, 580–596 (2012). [arXiv:1206.3418 [hep-ph]]

    Google Scholar 

  164. Blümlein, J., De Freitas, A., Van Hoeij, M., Imamoglu, E., Marquard, P., Schneider, C.: PoS LL 2018, 017 (2018). [arXiv:1807.05287 [hep-ph]]

    Google Scholar 

  165. Ronveaux, A. (ed.): Heun’s Differential Equations. The Clarendon Press Oxford, Oxford (1995)

    MATH  Google Scholar 

  166. Imamoglu, E., van Hoeij, M.: J. Symbolic Comput. 83, 245–271 (2017). [arXiv:1606.01576 [cs.SC]]

    Google Scholar 

  167. Takeuchi, K.: J. Fac. Sci Univ. Tokyo, Sect. 1A 24, 201–272 (1977)

    Google Scholar 

  168. Tricomi, F.G.: Elliptische Funktionen. Geest & Portig, Leipzig (1948). übersetzt und bearbeitet von M. Krafft

    Google Scholar 

  169. Whittaker, E.T., Watson, G.N.: A course of modern analysis. Cambridge University Press, Cambridge (1996). Reprint of 4th edition (1927)

    Google Scholar 

  170. Herfurtner, S.: Math. Ann. 291, 319–342 (1991)

    Article  MathSciNet  Google Scholar 

  171. Movasati, H., Reiter, S.: Bull. Braz. Math Soc. 43, 423–442 (2012). [arXiv: 0902.0760[math.AG]]

    Google Scholar 

  172. Blümlein, J.: Talks at: The 5th International Congress on Mathematical Software ZIB Berlin from July 11 to July 14, 2016, Session: Symbolic computation and elementary particle physics. https://www.risc.jku.at/conferences/ICMS2016/ (2016); and QCD@LHC2016, U. Zürich, August 22 to August 26, 2016. https://indico.cern.ch/event/516210/timetable/#all.detailed.

  173. Remiddi, E., Tancredi, L.: Nucl. Phys. B925, 212–251 (2017). [arXiv: 1709.03622[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1709.03622[hep-ph]]

    Google Scholar 

  174. Adams, L., Weinzierl, S.: Phys. Lett. B781, 270–278 (2018). [arXiv: 1802. 05020[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1802.05020[hep-ph]]

    Google Scholar 

  175. Serre, J.-P.: A Course in Arithmetic. Springer, Berlin (1973)

    Book  MATH  Google Scholar 

  176. Cohen, H., Strömberg, F.: Modular Forms, A Classical Approach. Graduate Studies in Mathematics, vol. 179. AMS, Providence (2017)

    Google Scholar 

  177. Ono, K.: The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series. CBMS Regional Conference Series in Mathematics, vol. 102. AMS, Providence (2004)

    Google Scholar 

  178. Chan, H.H., Zudilin, W.: Mathematika 56, 107–117 (2010)

    Article  MathSciNet  Google Scholar 

  179. Broadhurst, D.J.: Eta quotients, Eichler integrals and L-series, talk at HMI Bonn, February (2018)

    Google Scholar 

  180. Brödel, J., Duhr, C., Dulat, F., Penante, B., Tancredi, L.: JHEP 1808, 014 (2018). [arXiv:1803.10256 [hep-th]]

    Article  Google Scholar 

  181. Adams, L., Weinzierl, S.: Commun. Numer. Theor. Phys. 12, 193–251 (2018). [arXiv:1704.08895 [hep-ph]]

    Google Scholar 

  182. Bogner, C., Schweitzer, A., Weinzierl, S.: Nucl. Phys. B 922, 528–550 (2017). [arXiv:1705.08952 [hep-ph]]

    Google Scholar 

  183. Ablinger, J., Blümlein, J., De Freitas, A., Goedicke, A., Schneider, C., Schönwald, K.: Nucl. Phys. B 932, 129–240 (2018). [arXiv:1804.02226 [hep-ph]]

    Google Scholar 

  184. Blümlein, J. De Freitas, A., Raab, C., Schönwald, K.: The O(α 2) Initial State QED Corrections to \(e^+e^- \rightarrow \gamma ^*/Z_0^*\), [arXiv:2003.14289 [hep-ph]].

    Google Scholar 

  185. Ablinger, J., Blümlein, J., De Freitas, A., Schönwald, K.: Subleading Logarithmic QED Initial State Corrections to e +e −→ γ ∗∕Z 0 ∗ to O(α 6L 5), [arXiv:2004.04287 [hep-ph]].

    Google Scholar 

  186. Blümlein, J., De Freitas, A., Raab, C.G., Schönwald, K.: Phys. Lett. B 791, 206–209 (2019). [arXiv:1901.08018 [hep-ph]]

    Google Scholar 

  187. Blümlein, J., De Freitas, A., Raab, C.G., Schönwald, K.: Nucl. Phys. B 945, 114659 (2019). arXiv:1903.06155 [hep-ph]

    Google Scholar 

  188. Blümlein, J., Raab, C.G., Schönwald, K.: Nucl. Phys. B 948, 114736 (2019). arXiv:1904.08911 [hep-ph]

    Google Scholar 

  189. Besier, M., Van Straten, D., Weinzierl, S.: Commun. Num. Theor. Phys. 13, 253–297 (2018). arXiv:1809.10983 [hep-th]

    Google Scholar 

  190. Raab, C.G.: On the arithmetic of d’Alembertian functions (in preparation)

    Google Scholar 

  191. Guo, L., Regensburger, G., Rosenkranz, M.: J. Pure Appl. Algebra 218, 456–473 (2014)

    Article  MathSciNet  Google Scholar 

  192. Ablinger, J., Blümlein, J., Raab, C.G., Schneider, C.: J. Math. Phys. 55, 112301 (2014). [arXiv: 1407.1822[hep-th]]. http://www.arXiv.org/abs/[arXiv:1407.1822[hep-th]]

    Article  MathSciNet  Google Scholar 

  193. Hoffman, M.E.: J. Algebraic Combin. 11, 49–68 (2000). [arXiv:math/9907173 [math.QA]]

    Google Scholar 

  194. Blümlein, J.: Comput. Phys. Commun. 159, 19–54 (2004). [hep-ph/0311046]. http://www.arXiv.org/abs/[hep-ph/0311046]

    Google Scholar 

  195. Vermaseren, J.: Int. J. Mod. Phys. A14, 2037–2076 (1999). [hep-ph/9806280]. http://www.arXiv.org/abs/[hep-ph/9806280]

    Google Scholar 

  196. Blümlein, J., Kurth, S.: Phys. Rev. D60, 014018 (1999). [hep-ph/9810241]. http://www.arXiv.org/abs/[hep-ph/9810241]

    Google Scholar 

  197. Ablinger, J., Blümlein, J., Schneider, C.: J. Math. Phys. 52, 102301 (2011). [arXiv: 1105.6063[math-ph]]. http://www.arXiv.org/abs/[arXiv:1105.6063[math-ph]]

    Article  MathSciNet  Google Scholar 

  198. Ablinger, J., Blümlein, J., Schneider, C.: J. Math. Phys. 54, 082301 (2013). [arXiv: 1302.0378[math-ph]]. http://www.arXiv.org/abs/[arXiv:1302.0378[math-ph]]

    Article  MathSciNet  Google Scholar 

  199. Davydychev, A.I., Kalmykov, M.Yu.: Nucl. Phys. B699, 3–64 (2004). [hep-th/0303162]. http://www.arXiv.org/abs/[hep-th/0303162]

    Article  Google Scholar 

  200. Weinzierl, S.: J. Math. Phys. 45, 2656–2673 (2004). [hep-ph/0402131]. http://www.arXiv.org/abs/[hep-ph/0402131]

    Google Scholar 

  201. Reutenauer, C.: Free Lie Algebras. Calendron Press, Oxford (1993)

    MATH  Google Scholar 

  202. Nielsen, N.: Nova Acta Leopold. XC(3), 125–211 (1909)

    Google Scholar 

  203. Kölbig, K.S., Mignoco, J.A., Remiddi, E.: BIT 10, 38–74 (1970)

    Article  Google Scholar 

  204. Kölbig, K.S.: SIAM J. Math. Anal. 17, 1232–1258 (1986)

    Article  MathSciNet  Google Scholar 

  205. Devoto, A., Duke, D.W.: Riv. Nuovo Cim. 7N6, 1–39 (1984)

    Google Scholar 

  206. Lewin, L.: Dilogarithms and Associated Functions. Macdonald, London (1958)

    MATH  Google Scholar 

  207. Lewin, L.: Polylogarithms and Associated Functions. North Holland, New York (1981)

    MATH  Google Scholar 

  208. Remiddi, E., Vermaseren, J.A.M.: Int. J. Mod. Phys. A15, 725–754 (2000). [hep-ph/9905237]. http://www.arXiv.org/abs/[hep-ph/9905237]

    Google Scholar 

  209. Moch, S., Uwer, P., Weinzierl, S.: J. Math. Phys. 43, 3363–3386 (2002). [hep-ph/0110083]. http://www.arXiv.org/abs/[hep-ph/0110083]

    Google Scholar 

  210. Laporta, S.: Phys. Lett. B772, 232–238 (2017). [arXiv:1704.06996[hep-ph]]. http://www.arXiv.org/abs/[arXiv:1704.06996[hep-ph]]

    Article  Google Scholar 

  211. Blümlein, J., Vogt, A.: Phys. Rev. D 58, 014020 (1998). [hep-ph/9712546]

    Google Scholar 

  212. Blümlein, J., Ravindran, V., van Neerven, W.L.: Nucl. Phys. B 586, 349–381 (2000). [hep-ph/0004172]

    Google Scholar 

  213. Blümlein, J., Guffanti, A.: Nucl. Phys. Proc. Suppl. 152, 87–91 (2006). [hep-ph/0411110]

    Google Scholar 

  214. Blümlein, J.: Comput. Phys. Commun. 133, 76–104 (2000). [hep-ph/0003100]. http://www.arXiv.org/abs/[hep-ph/0003100]

    Google Scholar 

  215. Blümlein, J., Moch, S.-O.: Phys. Lett. B614, 53–61 (2005). [hep-ph/0503188]. http://www.arXiv.org/abs/[hep-ph/0503188]

    Google Scholar 

  216. Nielsen, N.: Handbuch der Theorie der Gammafunktion. Teubner, Leipzig (1906). Reprinted by (Chelsea Publishing Company, Bronx, New York, 1965)

    Google Scholar 

  217. Landau, E.: Über die Grundlagen der Theorie der Fakultätenreihen, S.-Ber. math.-naturw. Kl. Bayerische Akad. Wiss. München 36, 151–218 (1906)

    MATH  Google Scholar 

  218. Blümlein, J.: Clay Math. Proc. 12, 167–188 (2010). [arXiv:0901.0837[math-ph]]. http://www.arXiv.org/abs/[arXiv:0901.0837[math-ph]]

    MathSciNet  Google Scholar 

  219. Blümlein, J.: Comput. Phys. Commun. 180, 2218–2249 (2009). [arXiv:0901.3106[hep-ph]]. http://www.arXiv.org/abs/[arXiv:0901.3106[hep-ph]]

    Article  MathSciNet  Google Scholar 

  220. Kotikov, A.V., Velizhanin, V.N.: Analytic continuation of the Mellin moments of deep inelastic structure functions (2005). hep-ph/0501274

    Google Scholar 

  221. Ablinger, J.: PoS, 019 (2014). [arXiv:1407.6180[cs.SC]]. http://www.arXiv.org/abs/[arXiv:1407.6180[cs.SC]]

    Google Scholar 

  222. Ablinger, J.: A computer algebra toolbox for harmonic sums related to particle physics. Diploma Thesis, JKU Linz (2009). arXiv:1011.1176[math-ph]

    Google Scholar 

  223. Weinzierl, S.: Comput. Phys. Commun. 145, 357–370 (2002). [math-ph/0201011]. http://www.arXiv.org/abs/[math-ph/0201011]

    Google Scholar 

  224. Moch, S., Uwer, P.: Comput. Phys. Commun. 174, 759–770 (2006). [math-ph/0508008]. http://www.arXiv.org/abs/[math-ph/0508008]

    Google Scholar 

  225. Frellesvig, H.: Generalized Polylogarithms in Maple, arXiv:1806.02883 [hep-th].

    Google Scholar 

  226. Duhr, C., Dulat, F.: PolyLogTools – Polylogs for the masses, JHEP 08, 135 (2019). arXiv:1904.07279 [hep-th] H. Frellesvig, Generalized Polylogarithms in Maple, arXiv:1806.02883 [hep-th].

    Google Scholar 

  227. Gehrmann, T., Remiddi, E.: Comput. Phys. Commun. 141, 296–312 (2001). [hep-ph/0107173]. http://www.arXiv.org/abs/[hep-ph/0107173]

    Google Scholar 

  228. Maitre, D.: Comput. Phys. Commun. 174, 222–240 (2006). [hep-ph/0507152]. http://www.arXiv.org/abs/[hep-ph/0507152]

    Google Scholar 

  229. Ablinger, J., Blümlein, J., Round, M., Schneider, C.: PoS (RADCOR2017) 010 (2017). [arXiv:1712.08541[hep-th]]

    Google Scholar 

  230. Ablinger, J. Blümlein, J., Round M. Schneider, C.: %Numerical implementation of harmonic polylogarithms to weight w = 8, Comput. Phys. Commun. 240, 189–201 (2019). doi:10.1016/j.cpc.2019.02.005 [arXiv:1809.07084 [hep-ph]].

    Google Scholar 

  231. Vollinga, J., Weinzierl, S.: Comput. Phys. Commun. 167, 177 (2005). [hep-ph/0410259]. http://www.arXiv.org/abs/[hep-ph/0410259]

    Article  Google Scholar 

  232. Neumann, C.: Vorlesungen über Riemann’s Theorie der Abel’schen Integrale, 2nd edn. Teubner, Leipzig (1884)

    MATH  Google Scholar 

  233. Brown, F., Schnetz, O.: Duke Math. J. 161(10), 1817–1862 (2012)

    MathSciNet  Google Scholar 

  234. Hilbert, D.: Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. Teubner, Leipzig (1912)

    MATH  Google Scholar 

  235. Kronig, R. de L.: J. Opt. Soc. Am. 12, 547–557 (1926)

    Google Scholar 

  236. Kramers, H.A.: Atti Cong. Intern. Fisici, (Transactions of Volta Centenary Congress) Como 2, 545–557 (1927)

    Google Scholar 

  237. Abreu, S., Britto, R., Duhr, C., Gardi, E.: JHEP 12, 090 (2017). [arXiv:1704.07931[hep-th]]. http://www.arXiv.org/abs/[arXiv:1704.07931[hep-th]]

    Article  Google Scholar 

  238. Veltman, M.J.G.: Physica 29, 186–207 (1963)

    Article  MathSciNet  Google Scholar 

  239. Remiddi, E.: Helv. Phys. Acta 54, 364–382 (1982). Remiddi, E.: Differential equations and dispersion relations for feynman amplitudes. In: Blümlein J., Schneider C., Paule P. (eds.) Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, pp. 391–414. Springer, Wien (2019)

    Chapter  Google Scholar 

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Acknowledgements

I would like to thank J. Ablinger, D. Broadhurst, A. De Freitas, D. Kreimer, A. von Manteuffel, P. Marquard, S.-O. Moch, W.L. van Neerven, P. Paule, C. Schneider, K. Schönwald, J. Vermaseren and S. Weinzierl for countless fruitful discussions. This work was supported by the EU TMR network SAGEX Marie Skłodowska-Curie grant agreement No. 764850 and COST action CA16201: Unraveling new physics at the LHC through the precision frontier.

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    Johannes Blümlein

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    Carsten Schneider

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Blümlein, J. (2020). Large Scale Analytic Calculations in Quantum Field Theories. In: Pillwein, V., Schneider, C. (eds) Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-44559-1_5

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